Question

A small solid marble of mass $M$ and radius $r$ rolls down along the loop track, without slipping. Find the height $h$ above the base, from where it has to start rolling down the incline such that the sphere just completes the vertical circular loop or radius $R$.

Answer 1

To just complete the track potential energy at height h

(Point P)=(Kinetic+ potential) energy at H

$mgh=2mgR+\frac{1}{2}m{v}_0^2+\frac{1}{2}I{\omega }_0^2$

where $v_0$ is velocity required to complete the loop

$\frac{m{v}_0^2}{R}=mg\Rightarrow v_0^2=gR$

$mgh=2mgR+\frac{1}{2}m{v}_0^2+\frac{1}{2}\times \frac{2}{5}m{r}^2\times \frac{v_0^2}{R}$

$gh=2gR+\frac{1}{2}gR+\frac{1}{5}\frac{r^2}{R^2}.gR$ (${\omega }_0=\frac{v_0}{R}$ pure rolling)

$h=\frac{5R}{2}+\frac{R}{5}\left[\frac{r^2}{R^2}\right]$